Dafydd Evans

A proof of the Gamma test

From a dataset of input–output vectors, the Gamma test estimates the variance of the noise on an output modulo any smooth model with bounded partial derivatives. We present a proof of the Gamma test under fairly weak hypotheses.

Asymptotic Moments of Near-Neighbour Distance Distributions

Let C be a compact convex body in R⊃m and consider a set of points selected at random from C according to some well–behaved sampling distribution. We obtain an asymptotic expression for the positive moments of the kth near–neighbour distance distribution as the number of points increases to infinity.

A simple multi-objective optimization algorithm for the urban transit routing problem

The urban transit routing problem (UTRP) for public transport systems involves finding a set of efficient transit routes to meet customer demands. The UTRP is an NP-Hard, highly constrained, multi-objective problem, for which the evaluation of candidate route sets can prove both time consuming and challenging, with many potential solutions rejected on the grounds of infeasibility. In this paper we propose a simple evolutionary multi-objective optimization technique to solve the UTRP. First we present a representation of the UTRP and introduce our two key objectives, which are to minimise both passenger costs and operator costs. Following this, we describe a simple multi-objective optimization algorithm for the UTRP then present experimental results obtained using the Mandls benchmark data and a larger transport network.

A law of large numbers for nearest neighbour statistics

In practical data analysis, methods based on proximity (near-neighbour) relationships between sample points are important because these relations can be computed in time (n log n) as the number of points n→∞. Associated with such methods are a class of random variables defined to be functions of a given point and its nearest neighbours in the sample. If the sample points are independent and identically distributed, the associated random variables will also be identically distributed but not independent. Despite this, we show that random variables of this type satisfy a strong law of large numbers, in the sense that their sample means converge to their expected values almost surely as the number of sample points n→∞.

A computationally efficient estimator for mutual information

Mutual information quantifies the determinism that exists in a relationship between random variables, and thus plays an important role in exploratory data analysis. We investigate a class of non-parametric estimators for mutual information, based on the nearest neighbour structure of observations in both the joint and marginal spaces. Unless both marginal spaces are one-dimensional, we demonstrate that a well-known estimator of this type can be computationally expensive under certain conditions, and propose a computationally efficient alternative that has a time complexity of order (N log N) as the number of observations N→∞.

Non-parametric estimation of residual moments and covariance

The aim of non-parametric regression is to model the behaviour of a response vector Y in terms of an explanatory vector X, based only on a finite set of empirical observations. This is usually performed under the additive hypothesis Y=f(X)+R, where f(X)=(Y|X) is the true regression function and R is the true residual variable. Subject to a Lipschitz condition on f, we propose new estimators for the moments (scalar response) and covariance (vector response) of the residual distribution, derive their asymptotic properties and discuss their application in practical data analysis.

Noise Estimation in Long-Range Matched-Filter Envelope Sonar Data

In sonar signal processing when selecting a threshold for detection, it is necessary to consider the noise in the signal to achieve the desired rates of detection and false alarm. The clutter component of this noise, caused by scattering from environmental features, is often a limiting factor. This is particularly the case when active sonar systems operate in shallow water. Therefore, suitable modeling of clutter-limited data is vital for accurate detection in such environments. This paper investigates the K-distribution, the Weibull distribution, and the log-normal distribution as models for clutter-limited matched-filter envelope sonar data, obtained using FM chirp pulses in a shallow-water environment. The models are evaluated using modified Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) tests. Critical values for the upper tail AD statistic applied to the if-distribution are estimated by Monte Carlo simulation and tabulated here. Results show that the if-distribution and the Weibull distribution provide a good model of noise in clutter-limited environments. However, the K-distribution provides a better fit in the tails, which is important for target detection. The Kolmogorov-Smirnov test is shown to be an unsuitable method of evaluating fit when the tail of a distribution is of greatest interest. We also show that the estimated shape parameter of the K-distribution does provide a simple means of identifying regions dominated by clutter.

Estimating the Variance of Multiplicative Noise

When constructing non‐parametric models from noisy data, it is useful to have information regarding the statistical properties of the noise distribution. In many cases, such information is not explicitly available, and must be estimated directly from the data. Under the hypothesis of additive noise, algorithms for estimating the variance of the noise distribution have appeared in the literature. In this paper we present a novel algorithm for estimating the noise variance under a multiplicative hypothesis.

Difference-based Methods for Truncating the Singular Value Decomposition

Given a noisy time series (or signal), one may wish to remove the noise from the observed series. Assuming that the noise-free series lies in some low-dimensional subspace of rank r, a common approach is to embed the noisy time series into a Hankel trajectory matrix. The singular value decomposition is then used to deconstruct the Hankel matrix into a sum of rank-one components. We wish to demonstrate that there may be some potential in using difference-based methods of the observed series in order to provide guidance regarding the separation of the noise from the signal, and to estimate the rank of the low-dimensional subspace in which the true signal is assumed to lie.

Manage the emotional needs of learners in teaching sessions

Emotions influence learning by affecting our thinking, behaviour and memory. Emotional learning attempts to create a caring and engaging learning environment. Giving appropriate consideration to and managing emotions during educational encounters can improve interest and attention, reasoning, problem solving and decision-making in learners as well as motivation and energy in the teacher. Emotionally intelligent educators will be able to generate and use emotions, where appropriate, to help learners achieve their learning outcomes. In addition, they may have a better rapport with learners, enhance emotional health and improve academic and work performance.[1] If strong emotions are generated during or after a teaching session, the memories related to this will have a strong emotional component. Emotional memory is said to be long lasting. If it is linked to sensory input – sound, smell or visual items – the learners may experience a resurgence of the same emotions when the same stimulus is re-applied. There are five core emotions – happiness, sadness, surprise, disgust and fear. Sometimes learners’ emotional states can be deliberately manipulated in a variety of ways, e.g. using music, games, stories, films, etc. to enhance the learning. However, it is not uncommon for difficult emotions to be triggered unintentionally and unexpectedly. Some of the emotions could be disruptive to the learning and some could be beneficial. Good teaching and facilitation involves managing and dealing with emotions to enhance learning but also to ensure the psychological well-being of the learner. It is the latter that is paramount. On rare occasions where emotional reactions are more severe, learners may need additional short-term support. It is essential when planning a session that the teacher is able to foresee the potential for strong emotional reactions, including unexpected responses, and ensure that measures to tackle these situations are in place.